Powerfree integers and Fourier bounds

We develop a general approach for showing when a set of integers $\mathscr{A}$ has infinitely many $k^{th}$ powerfree numbers. In particular, we show that if the Fourier transform of $\mathscr{A}$ satisfies certain $L^{\infty}$ and $L^{1}$ bounds, and is also ``decreasing'' in some sense, then $\mathscr{A}$ contains infinitely many $k^{th}$ powerfree numbers. We then use this method to show that there are infinitely many cubefree palindromes in base $b$ for $b\ge 1100$.